The home secretary, Theresa May, has unveiled the coalition's compromise package on the student visa system and claimed it would curb numbers by more than 25%, with up to 80,000 fewer coming to Britain each year.

May claimed that new restrictions on the ability of overseas students to stay on and work for up to two years on graduation would reduce numbers by a further 20,000.

The home secretary said the curb was needed not only to tackle abuse but to meet the government's target of reducing annual net migration to below 100,000 a year: "The most significant migrant route to Britain is the student route. And so we must take action here too," she saidon Tuesday, adding that together with their dependants, students accounted for two-thirds of all visas issued to non-EU migrants last year.

The final package to come into force from April involves tighter rules on the accreditation of private colleges that can sponsor overseas students; tougher English language requirements; tighter rules on students working part-time; restrictions on which students can bring dependants with them; maximum time limits of between three and five years on the length of courses; and changes to the post-study work route.

It is understood that Whitehall negotiations over the final detail of the delayed package went on until late on Monday, with May giving way on some key details.

Liberal Democrat sources said that Nick Clegg and Vince Cable had successfully battled to prevent the Tories imposing an arbitrary cap on student migration and had forced the Home Office to change "potentially devastating" plans.

"Nick and Vince were absolutely adamant about this," said a senior Liberal Democrat source. "The last thing we need to be doing when we are encouraging growth is to pull the rug out from under the feet of our great universities. The draconian restrictions originally planned by the Home Office would have been a slap in the face for British colleges, universities and businesses. This is a victory for reason over prejudice."

The universities, English-language schools and the National Union of Students acknowledged that the government had responded to their concerns in its final package. Universities UK said the proposals would allow the £5bn annual contribution by international students in tuition fees and off-campus spending to continue.

But the shadow home secretary, Yvette Cooper, claimed May was simply restricting student visas in one area while allowing them to rise in a separate short-term student visitor category which does not count towards the net migration target.

"Isn't the real truth that this policy is not about youth unemployment or bogus courses, it is about hitting higher education because you can't meet your promise to cut migration to tens of thousands over the course of this parliament?"

The issues which the Liberal Democrats claimed progress on included:

• Pathway courses: proposals to bar overseas students coming to further education colleges to learn English before going to university or public school have been dropped as long the main course provider sponsors the student directly.

• Post-study work route: a possible option was to abolish or cap the ability of students to seek work for two years on graduation from university. Now graduates who have the offer of a graduate-level skilled job paying at least £20,000 a year will be able to stay on to work. However a cap on these numbers will be kept in reserve if they start to rise sharply. The numbers of "switchers" will not count to the overall limit on net migration.

• Increased English-language requirement: the standard is to be raised to intermediate proficiency as defined by the secure English language test, but universities will be exempted from this requirement and be able to impose their own test.

The bulk of the 70,000 to 80,000 reduction in the 262,000 student visas issued each year is expected to come from the changes in accrediting private colleges in the higher and further education sector. Currently only 113 out of the 700-odd such colleges enjoy "highly trusted sponsor" status with the Home Office. A new licensing system will determine the fate of the 600 other colleges who rely on overseas students.

Universities UK said the final package would allow British universities to remain at the forefront of international student recruitment.

However the English-language schools warned that a new loophole could be opened up through the extended student visitor visa which allows people to come for up to 11 months and does not count to the overall figures on net migration.